Modeling non-Fickian pollutant mixing in open channel flows using two-dimensional particle dispersion model
Introduction
In streams and rivers, in the intermediate mixing region after the completion of vertical mixing, pollutant mixing can be modeled by adopting the two-dimensional (2D) dispersion theory, as shown in Fig. 1. This mixing process is called ‘shear flow dispersion’, in which, for 2D shear flow shown in Fig. 1b) and c), the pollutant column is stretched due to the different velocities over the depth in both the longitudinal (s) and transverse (n) directions. The stretched pollutant column is then well mixed due to the turbulent diffusion in the vertical direction after time, t. Finally, the pollutant cloud is widely dispersed, and the depth-averaged concentration curves spread wider with time, as shown in Fig. 1d. Among the various mixing mechanisms affecting the mixing of pollutant slugs that are introduced into the streams and rivers, the dispersion of the pollutant cloud by the shear flow is regarded as the most important because the shear flow is the largest fluctuation that controls the motion of water parcels over the whole flow field of the river (Daily and Harleman, 1966, Schwab and Rehmann, 2015). In order to model the aforementioned dispersion mechanism using a 2D approach, a 2D advection-dispersion equation has previously been used (Fischer et al., 1979, Rutherford, 1994). This equation can be derived by averaging the three-dimensional (3D) diffusion equation for turbulent flows over the depth (Rutherford, 1994) in the natural coordinate system aswhere z is the vertical direction; h is the water depth; εL and εT are the longitudinal and transverse turbulent diffusion coefficients, respectively; and are the depth-averaged velocity components corresponding to the s- and n-directions, respectively; ; us and un are the time-averaged velocity components which vary over depth; C is the depth-averaged concentration; c′ = c − C; and c is the time-averaged concentration. As aforementioned, the effect of the shear flow on the 2D mixing is incorporated into the correlation term on the right-hand side of Eq. (1), and this dispersive mass flux is due to the combined effect of shear transport and vertical diffusion as shown in Fig. 1.
To model the mass flux term on the right hand side of Eq. (1), Fischer (1978) obtained the analytical solution of c′(z) by simply expanding the one-dimensional (1D) dispersion theory originally introduced by Taylor (1954) to 2D mixing, then found the relation as follows:where DL and DT are the longitudinal and transverse dispersion coefficients, respectively. This result concurs with the Fickian diffusion for molecular diffusion. This model is therefore called the 2D Fickian dispersion model because the mass transport due to the combined effect of shear advection and vertical diffusion follows Fick's law. Further, in Eq. (2), the combined effect of shear flow and turbulent diffusion is incorporated into the single dispersion coefficient, which is calculated by the following equation (Fischer, 1978).where εV is the vertical turbulent diffusion coefficient. As aforementioned, the dispersion coefficient in Eq. (3) is the bulk mixing coefficient which integrates the shear effects of the velocity variation over the entire flow depth, and thus is usually much larger than the turbulent diffusion coefficients (Daily and Harleman, 1966).
Finally, the 2D Fickian dispersion model can be derived by substituting Eq. (2) into Eq. (1) as follows.
The Fickian dispersion model, Eq. (4), has previously been used to analyze the 2D pollutant mixing problems both in the Eulerian (Piasecki and Katopodes, 1999, King et al., 2005, Lee and Seo, 2007,Lee and Seo, 2010, Seo et al., 2008,2010) and Lagrangian frames (Dimou and Adams, 1993, Weitbrecht et al., 2004, Suh, 2006). However, two major limitations arise in using the Fickian dispersion model. Firstly, the 2D Fickian dispersion model demands the input of both longitudinal and transverse dispersion coefficients, which are the most important parameters of the model. However, it is difficult to estimate these coefficients using Eq. (3) because the formula requires the vertical profiles of the longitudinal and transverse velocities which are not easily measured in the laboratory and field experiments. Thus, a number of tracer studies have previously been conducted to find the 2D dispersion coefficients. The transverse dispersion coefficient (DT) was obtained from the field tests using the continuous injection of tracers (Yotsukura et al., 1970, Holley and Abraham, 1973, Jackman and Yotsukura, 1977, Sayre, 1979, Beltaos, 1980, Lau and Krishnappan, 1981, Holly and Nerat, 1983, Rutherford, 1994, Boxall and Guymer, 2003, Zhang and Zhu, 2011), while both DL and DT were obtained from the 2D tracer tests with instantaneously introduced tracers (Seo et al., 2006,2016; Baek and Seo, 2010). However, despite the many field and laboratory studies, it is still difficult to determine the dispersion coefficients from the measured concentration data since these coefficients strongly depend on the various hydraulic and geometric parameters of the river.
Secondly, the 2D Fickian dispersion model has an inherent limitation since the 2D Fickian dispersion model was derived based on the Taylor's 1D dispersion theory. Thus, as this model applied the same assumption that the balance between shear advection in the horizontal direction and the turbulent diffusion in the vertical direction is successfully achieved, this model can be applied only after the so-called initial period after the injection, given by Chatwin (1970) aswhere tI is the initial period. Thus, as presented by Chatwin (1970), up to this time, the concentration curves are asymmetrical, while after sufficient time, the so-called Taylor period, the curves approach the symmetric distribution, which would be predictable by the Fickian dispersion model. Therefore, in the initial period, an alternative method to the Fickian dispersion model to calculate the pollutant mixing is required, instead of using Fick's law.
To overcome these two limitations of Taylor's model, non-Fickian dispersion models that avoid the use of Fick's law have been suggested. Aris (1956), using the concentration moment method, obtained the same results of the 1D Fickian dispersion model without introducing the assumption by Taylor (1954). Thus, Aris's theory can be applied to both the initial and Taylor periods, even though model equations are expressed via the moment of the concentration distribution. Chatwin (1970) also derived an asymptotic series solution based on Aris's theory to analyze the asymmetric concentration curves in the initial period, which was suggested as Eq. (5).
To compute the concentrations in both the initial and Taylor reaches, Fischer (1968) presented a step-by-step calculation method which calculates the shear advection and transverse diffusion steps sequentially. The pollutant mixing in the initial period was therefore able to be analyzed without using Taylor's theory. The 1D step-by-step calculation method divides the continuous solute mixing procedure into discontinuous calculation steps, which consist of the longitudinal shear advection (Eq. (6a)) and the lateral mixing (Eq. (6b)), according to the physical interpretation of the 1D shear dispersion as shown in Fig. 2.where I is the index of longitudinal grid; J is the jth stream tube; HV is the Heaviside step function (which equals + 1 if the argument is positive and zero otherwise); is the dimensionless convective velocity; t1 is the time after the longitudinal advection; Δt is the time step; and kt is the transfer coefficient between adjacent stream tubes. In this method, the pollutant cloud is transported to the downstream computational grid in the shear advection step, and the concentration is then diffused to the next grid in the lateral mixing step. Fischer (1968) applied this method to simulate the Green-Duwamish River using eight stream tubes. Based on the computation results of two different sources, a line source and a point source, he maintained that the effect of source type on the longitudinal variance is small.
The step-by-step calculation method was revisited by Seo and Son (2006) to simulate 1D longitudinal pollutant mixing in straight open channel flows in which they used empirical equations for a lateral velocity profile. Jung and Seo (2013) was the first to adopt the step-by-step method for simulating 2D pollutant mixing in open channels, in which they applied theoretical equations for the vertical velocity profiles by Rozovskii (1957) and Odgaard (1986). The equations for the sequential calculation to represent the 2D dispersion processes read as follows:where s1 and n1 are specific positions in longitudinal and transverse directions, respectively; L is the number of horizontal planes over the water depth. Eqs. (7a) and (7c) describe the horizontal translations of the pollutant cloud caused by the skewed vertical velocity profiles. The vertically well-mixed pollutant cloud is then calculated using Eqs. (7b) and (7d). Jung and Seo (2013) maintained that their 2D model could generate the skewed concentration curves in the initial period and the symmetrical curves in the Taylor period.
The aforementioned non-Fickian dispersion models based on the step-by-step calculation of shear advection and turbulent diffusion can be classified as the Lagrangian method because the water parcels containing the pollutant mass are transported to the next computational grid by the velocity of each grid. Thus, in these models, advection due to shear flow can be adequately described, even though random fluctuation by turbulent motion was not included. However, tracking of the polluted grid rather than tracking the individual pollutant particle would induce large computation errors due to the grid resolution. This so-called numerical diffusion error was defined by Fischer (1968) as a function of time step and grid size, and he suggested that the grid size should be small to ensure the effect of numerical diffusion remained insignificant compared to the physical dispersion. Thus, in order to avoid this numerical error, a particle tracking method in which random properties of pollutant transport can be modeled more effectively needs to be developed.
In this study, the 2D particle dispersion model was developed to model the 2D pollutant mixing in both the initial and Taylor periods without adopting Fick's law. The proposed model describes the physical process of shear dispersion through sequential calculations of the horizontal transport and vertical mixing of pollutant particles as an alternative to tracking the polluted grid in which particles are transported by both deterministic velocity component and random fluctuations. The simulation results of the proposed model were compared with the Fickian dispersion model and experimental results in the straight and meandering channels during the initial and Taylor periods. Furthermore, the applicability of the proposed model was tested against the tracer results obtained in the Hongcheon River, South Korea.
Section snippets
Model development
In this study, the non-Fickian dispersion model based on the step-by-step calculation procedure was developed using the Lagrangian particle tracking method. In the surface water modeling, the particle tracking method has been used for the simulation of various environmental problems, such as prediction of larva migration, dilution of heated water, and transport of miscible and immiscible pollutants (Kitanidis, 1994, Pedersen et al., 2003, Chang et al., 2011, Hyun et al., 2012). The non-Fickian
Mixing in the initial and Taylor periods
In this study, pollutant mixing in the straight open channel flow was analyzed using 2D PDM in both the initial and Taylor periods. In this study, flow in the straight channel was assumed to be the plane shear flow, in which the vertical profile of longitudinal velocity has a logarithmic profile and the effect of transverse shear flow is negligible (Rutherford, 1994). In the numerical modeling, pollutant mixing simulations were conducted in various flow conditions in which the cross-sectional
Meandering channel
The pollutant transport simulations using the 2D PDM were conducted in the meandering channel, which was depicted in Fig. 10. The simulation results were then compared with the tracer test results by Seo and Park (2009) in which the salt solution was introduced instantaneously as the vertical line source. The meandering channel has a rectangular cross-section with a channel width of 1 m, and the two channel bends were connected with the straight part with a length of 1 m. The experimental and
Conclusions
In this study, the 2D particle dispersion model, 2D PDM, was developed to model the 2D pollutant mixing in open channel flows. Instead of using the Taylor's approach, the 2D PDM adopted the step-by-step arithmetic calculation method to reproduce shear dispersion by the vertical variations of the longitudinal and transverse velocities. The calculation procedures of 2D PDM included the horizontal transport and vertical mixing steps. In the horizontal transport step, the introduced pollutant
Acknowledgments
This research was supported by a Grant (11-TI-C06) from Advanced Water Management Research Program funded by Ministry of Land, Infrastructure and Transport of Korean Government. This work was conducted at the Research Institute of Engineering and Entrepreneurship and the Institute of Construction and Environmental Engineering in Seoul National University, Seoul, Korea.
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